Ask the Wizard #366

Here is my solution (PDF).

Let's first answer the question of what is the probability the player will be down more than x units after one million flips, assuming the player has an unlimited bankroll.

Since this is a fair bet, the mean win after a million flips is zero. The variance of each flip is 1, so the variance of one million flips is one million. One standard deviation is thus sqrt(1,000,000) = 1000.

We can find the bankroll required with the Excel function =norm.inv(probability,mean,standard deviation). For example, if we put in =norm.inv(.25,0,1000), we get -674.49. This means if after one million flips, the player has a 25% chance of being down 674 or more. Please keep in mind this is an estimate. To get a true answer, we should use the binomial distribution, which would be very tedious with a million flips.

It could very well happen that if the player took $674 to the table, he might run out of money before the million flips. If he could keep playing on credit, it might happen that he has a recovery and finishes less than $674 down. In fact, once the player is at -674, there is a 50/50 chance he will end up above or below -674 at any given point in the future.

So, if the player can play on credit, there are three possible outcomes.

1. Player never falls below -674.

2. Player falls below -674 at some point, but recovers and finishes above -674.
3. Player falls below -674 at some point, keeps playing and loses even more.

We have established scenario 3 has a probability of 25%.

Scenario 2 must have the same probability as scenario 3, because once the player is down -674, he has a 50/50 chance to finish above or below that point after one million flips.

Scenario 1 is the only other alternative, which must have probability 100%-25%-25% = 50%.

If the probability the player never falls below 674 is 50%, then the alternative of falling below must be 100%-50% = 50%.

So, there is our answer to the original question, $674.

This question is asked and discussed in my forum at Wizard of Vegas.

This question is asked and discussed in my forum at Wizard of Vegas.